# Matrix factorization method

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matrix forward-backward substitution method

A method for solving finite-difference systems that approximate boundary value problems for systems of ordinary differential equations in one-dimensional problems, and for elliptic equations in two-dimensional problems.

The solution of the three-point difference scheme where is an unknown grid vector, is the right-hand side vector and are given square matrices, under the boundary conditions is sought for, as in the scalar case, in the form (*)

The coefficients (the matrix and the vector ) are determined from the recurrence relations ( "forward substitution" )  while and are given by the left boundary condition: The are calculated by formula (*) ( "backward substitution" ), and There is stability of this method to rounding errors under the conditions  which implies that , (see ). A different form of the stability conditions is also available (see , ). The matrix factorization method is applied also to two-point difference schemes (see ). A variant in which inversion of matrices is replaced by orthogonalization is also used (see ).

How to Cite This Entry:
Matrix factorization method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_factorization_method&oldid=15766
This article was adapted from an original article by T.A. Germogenova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article